Home > Uncategorized > A Model of All You Can Eat

A Model of All You Can Eat

Frances Wooley has a post about all-you-can-eat sushi.  It was meant to illustrate a broad, overarching truth about economics.  That point is valid, but the discussion got a little sidetracked by the al-you-can-eat-sushi issue.  Since this is the kind of thing I am actually trained in (as opposed to monetary economics which I am basically absorbing/making up as I go along), I figure I will interject myself.  I tried to explain it in the comments but I need a bit more space to really knock it out of the park.  Here it goes.

Consider a market for restaurants in which there are two costs of production which are both variable but one is more variable than the other.  One cost I will call “tables” but it can represent parking spaces, floor space, some component of cooks, wait staff and ovens and anything that is required to serve more customers.  The other I will call “food” and it represents whatever is required to serve a given customer additional food.  This may include some component of things like cooks and ovens and wait staff but likely not all.  But just say -they buy tables and food.

What the customers want is a table and some quantity of food.  The amount of “tables” that a given number of customers use is fixed but the amount of food depends on the price.  Thus, they don’t care about the table, they just need it to eat the food.  So we can characterize their demand with just a demand for food.  Let there be two types of customers, heavy eaters and light eaters and let their respective demands be the following.

Dh=10-P

Dl=5-P

Where P is the price per unit of food.  Also assume that the population of customers is half heavy and half light eaters and restaurant owners can’t tell the difference (or at least can’t accept only one kind and not the other).

Finally, assume that the restaurants are price-takers in the markets for food (let the cost of food be C) but face a u-shaped cost curve for tables and assume that the market for customers is perfectly competitive.   So in equilibrium, all restaurants have to offer the same consumer surplus to customers of a given type and they will expand to the scale (in terms of customers) for which the marginal cost of tables is equal to the average cost of tables and the marginal revenue (given the optimal pricing scheme, whatever that is) per table is equal to the average cost of tables.

Now consider how food is priced.  If restaurants can charge a fixed price for a table and then a separate price for food, the perfectly competitive equilibrium would be for the price of a table to equal the average (and marginal) cost of a table and the price of food to equal C.  But let’s imagine that this is not possible and the restaurants can only charge a single price for food.  In this case, that price cannot be C because they would all lose money.  Competition can only push it down to the point where the average profit per customer is equal to the average cost per table.

Average profit per customer will be given by the following expression.

1/2(10-P)(P-C)+1/2(5-P)(P-C)

If we assume C=1 and simplify we get:

8.5P-P^2-7.5

Now if we assume that the average cost of a table is 9, we can set this equal to the above expression and solve for the “competitive” price of food, which will be 3.  Notice that this is higher than the marginal cost of food but that the firm cannot do any better without deviating from the efficient scale.  If they add another table it will cost them more than 9 and the average increase in revenue will only be 9 at the market price.  And even though they would be willing to sell another unit of food to each customer at a price lower than 3, if they lowered the price on all units, their profit would become negative and they would go out of business.

However, notice that the marginal benefit for each type of eater is not the same because the heavy eaters buy more food, and the price of food is higher than the marginal cost.  At the market price of 3, the light eaters buy 2 units of food and the heavy eaters buy 7.  This means that heavy eaters yield a profit of 7x(3-1)-9=5 while light eaters yield 2(3-2)-9=-5.  So if the restaurant could attract more heavy eaters and fewer light eaters, they would profit more.  This means that they would be willing to charge a lower average price to a heavy eater.  This can be accomplished with an “all-you-can-eat” pricing scheme.

Let’s assume that all other restaurants practice standard pricing and the market price is 3.  If you offered “all you can eat” meaning that the marginal price of food is zero, a heavy eater would eat 10 units.  This is three units more than with standard pricing (though only one unit more than the efficient quantity since standard pricing leads to underconsumption of food).  Their consumer surplus from another restaurant would (calculated the econ 101 way) would be 7+6+5+4+3+2+1=28 and their total value of the ten units would be 55 so they would be willing to take the all you can eat option for up to 27.  This would mean that your profit from heavy eaters would be 27-10×1-9=8.

What about the light eaters?  A light eater, facing a zero marginal price of food would buy 5 units for a total value of 15.  So if you charge 27, you will get no light eaters.  This means that your average profit per customer will be 8 which is greater than zero so the all-you-can-eat pricing is better.

So there you have it.  The nature of the inputs and the restriction on the price structure (no two-part pricing) makes some inefficiency inevitable.  The two types of customer means that by selecting a single type, you can trade off a different type of inefficiency in a more efficient manner.  Of course, this will mean that an equilibrium will have some of each with all heavy eaters going to all you can eat places and the price there will be driven down more while all light eaters will go to standard pricing places and the price there will actually go up (there could be a pooling all you can eat equilibrium but I am too lazy to check) but this should show how such a thing is feasible.  I’m also too lazy to edit this so there might be a slight error here or there but I think the overall logic will stand up.

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Categories: Uncategorized
  1. Tom Brown
    June 20, 2014 at 5:24 pm

    Mike, wasn’t that Frances Woolley’s article about the Sushi?

    • Tom Brown
      June 20, 2014 at 6:00 pm

      The by-line at the bottom:
      “Posted by Frances Woolley on June 18, 2014 in Everyday economics, Frances Woolley | Permalink “

    • Free Radical
      June 20, 2014 at 6:22 pm

      Yes, my mistake. Thanks.

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